Question: In triangle $ABC$, altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$.  If $\angle ABC = 49^\circ$ and $\angle ACB = 12^\circ$, then find the measure of $\angle BHC$, in degrees.
Explanation: Note that triangle $ABC$ is obtuse, so $H$ lies outside triangle $ABC$.

[asy]
unitsize(1 cm);

pair A, B, C, D, E, F, H;

B = (0,0);
C = (4,0);
A = extension(B, B + dir(49), C, C + dir(180 - 12));
D = (A + reflect(B,C)*(A))/2;
E = (B + reflect(C,A)*(B))/2;
F = (C + reflect(A,B)*(C))/2;
H = extension(B,E,C,F);

draw(B--H--C--cycle);
draw(H--D);
draw(B--F);
draw(C--E);

label("$A$", A, SE);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, S);
label("$E$", E, W);
label("$F$", F, NE);
label("$H$", H, N);
[/asy]

Since triangle $BEC$ is right, $\angle CBE = 90^\circ - \angle BCE = 90^\circ - 12^\circ = 78^\circ$.  Since triangle $BFC$ is right, $\angle BCF = 90^\circ - \angle CBF = 90^\circ - 49^\circ = 41^\circ$.  Therefore, $\angle BHC = 180^\circ - \angle CBH - \angle BCH = 180^\circ - 78^\circ - 41^\circ = \boxed{61^\circ}$.